To tackle this problem, we need to apply the principles of the photoelectric effect, which describes how light can eject electrons from a metal surface. The key to solving this is understanding the relationship between the energy of the incident photons, the work function of the metal, and the kinetic energy of the emitted photoelectrons. Let's break it down step by step.

Understanding the Photoelectric Effect

The energy of a photon can be calculated using the formula:

E = h * f

where E is the energy of the photon, h is Planck's constant, and f is the frequency of the light. Since frequency and wavelength are related by the equation c = f * λ (where c is the speed of light and λ is the wavelength), we can also express the energy in terms of wavelength:

E = (h * c) / λ

Given Data

• Maximum K.E of photoelectrons for wavelength λ1: 30 eV
• Maximum K.E of photoelectrons for wavelength λ2: 10 eV
• Planck's constant (h): 4.14 x 10^-15 eV-s
• Speed of light (c): 3 x 10⁸ m/s

Calculating the Energies

From the photoelectric effect, we know that:

E = Work Function + K.E

Let’s denote the work function of the metal as Φ.

For the first wavelength (λ1):

E1 = Φ + K.E1

E1 = Φ + 30 eV

For the second wavelength (λ2):

E2 = Φ + K.E2

E2 = Φ + 10 eV

Finding the Energies of the Photons

Now, we can express the energies in terms of the wavelengths:

E1 = (h * c) / λ1

E2 = (h * c) / λ2

Setting Up the Equations

Substituting these into our earlier equations gives us:

(h * c) / λ1 = Φ + 30

(h * c) / λ2 = Φ + 10

Subtracting the Equations

By subtracting the second equation from the first, we eliminate Φ:

(h * c) / λ1 - (h * c) / λ2 = 20

Factoring out h * c:

h * c * (1/λ1 - 1/λ2) = 20

Calculating λ1 and λ2

Now, we can rearrange this to find the difference in wavelengths:

1/λ1 - 1/λ2 = 20 / (h * c)

Substituting the values of h and c:

1/λ1 - 1/λ2 = 20 / (4.14 x 10^-15 * 3 x 10⁸)

Calculating the right side:

1/λ1 - 1/λ2 = 20 / (1.242 x 10^-6)

1/λ1 - 1/λ2 = 1.61 x 10⁶

Finding λ2

Now, we need to find λ2. We can use the second equation:

λ2 = (h * c) / (Φ + 10)

We can express Φ in terms of λ1:

Φ = (h * c) / λ1 - 30

Substituting this into the equation for λ2 gives us:

λ2 = (h * c) / ((h * c) / λ1 - 30 + 10)

Finding Maximum Wavelength

The maximum wavelength for which photoelectrons can be emitted corresponds to the work function being equal to the energy of the incident photon:

Φ = (h * c) / λ_max

Rearranging gives:

λ_max = (h * c) / Φ

Substituting the value of Φ we calculated earlier will yield the maximum wavelength.

Final Calculations

After performing the calculations, we find:

• λ1 ≈ 310.5 Å
• λ_max ≈ 1242 Å

This means that the maximum kinetic energy of photoelectrons emitted from the metallic surface is achieved with a wavelength of approximately 310.5 Å, and the maximum wavelength of incident radiation for which photoelectrons can still be emitted is about 1242 Å. This illustrates the fundamental principles of the photoelectric effect and the relationship between light and electron emission in metals.